How can I use checker deselect to select separated faces?
When converting between selection modes (vertex,Edge,Face), Blender follows these rules:


When converting to a higher order (V>E>F) the higher-order components become selected if all their lower-order components were selected.
When converting to a lower order (F>E>V) all the lower-order components of previously selected higher-order components become selected.


Making use of that behavior:


Select 2 sequential longitudinal edges in the pattern (from the first and second regions to be selected in the end).
Hit CtrlShiftNumpad+ (Select Next Element) repeatedly to continue selecting further edges at the same interval
From the header Select menu, hit 'Select Loops'> 'Edge Rings'
Switch to Vertex mode.


The faces/all their edges in the regions are now implicitly selected, if you switch back between modes.



In the picture, I switched to vertex mode before making the edge ring selection .. that still works (by the conversion rules)